Corollary 11.10.1.19. Let $q: X \rightarrow S$ and $q': X' \rightarrow S$ be cocartesian fibrations of simplicial sets which are equivalent (in the sense of Definition 11.10.1.10). Then $q$ is a left fibration if and only if $q'$ is a left fibration.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Combine Propositions 11.10.1.21 and 5.1.4.15. $\square$